190 problems

Gromov–Witten/Donaldson–Thomas correspondence. The GW theory and the DT theory of $X$ should be equivalent after a change of variables.

Let $Z_{PT}(q)$ denote the stable-pairs partition function, and let … be the weighted generating function of labelled box configurations, with boundary partitions $(,,)$. Pandharip…

The paper considers cohomological Donaldson–Thomas theory as a limit of the K-theoretic computations developed there. Donaldson–Thomas/Gromov–Witten correspondence. Cohomological D…

Let $X$ be a polarized Calabi–Yau $3$-fold, let $\overline{\Omega}_\gamma^X\in\mathbb{Q}$ be its Donaldson–Thomas invariant, and define the BPS invariant…

Bayer–Macrì–Toda Bogomolov–Gieseker inequality. For any $(b,w)\in U$ and $\nu_{b,w}$-semistable $E\in\mathcal D(X)$, one has

Let $X$ be a nonsingular projective $3$-fold. The Gromov–Witten theory of $X$ consists of its Gromov–Witten invariants, and the Donaldson–Thomas theory consists of its Donaldson–Th…

Degree-zero DT conjecture. The degree-zero Donaldson–Thomas partition function is

Let $(Y\mid\partial Y)$ be a logarithmic pair, with logarithmic Donaldson–Thomas and stable-pairs generating series, and let the degree-$0$ evaluation denote the degree-$0$ contrib…

Let $X$ be a smooth projective threefold with a smooth divisor $D$, let $\beta\in H_2(X,\mathbb{Z})$, and let $\eta$ be a relative partition. Write…

Let $S$ be a smooth projective surface, let $X=\operatorname{Tot}_S(\omega_S)$ be its associated local surface, and let $v=(\beta,n)\in N_{\leq 1}(S)$ be primitive with $\beta>0$.…

Let $\widetilde Q$ be a quiver with potential $\widetilde W$, let $G$ be the relevant group, and let $\upzeta$ be a stability condition. Write $\mathtt{JH}$ for the semisimplificat…

Independence conjecture. The series $\mathsf{DT}_{r,k}^{\rm ell}(\mathbb{A}^3,q,t,w;p)$ does not depend on the elliptic parameter $p$.

Let $t_1,t_2,t_3,t_4,y,q$ be formal variables satisfying $t_1t_2t_3t_4=1$. For any formal variable $x$, write $[x]=x^{\frac12}-x^{-\frac12}$, and let $\operatorname{Exp}$ denote th…

Let $S$ be a K3 surface with symplectic involution $g$, let $X=(S\times E)/\mathbb{Z}_2$, and let $\beta=(\gamma,d)\in H_2(X,\mathbb{Z})$ be a curve class. Define…

Katz's genus-zero GV/DT conjecture.

Genus-zero GW/DT4 conjecture. For a suitable choice of orientation,

Let $\mathcal{Z}$ be a hard-Lefschetz $3$-orbifold, and let … be a crepant resolution. Let $DT_{mr}(-)$ denote the reduced, multi-regular Donaldson–Thomas potential, and let…

Let $X$ be a Calabi–Yau threefold. Write $\widetilde{N}_{\chi,\beta}$ for Donaldson–Thomas invariants counting ideal sheaves with holomorphic Euler characteristic $\chi$ and curve…

MNOP rationality conjecture. The function $Z'_{DT}(\boldsymbol{\gamma};q)_\beta$ is a rational function of $q$.

Let $X$ be a smooth quasi-projective $3$-fold over $\mathbb{C}$ equipped with an action of an algebraic torus $T$, and suppose that the fixed locus $X^T$ is compact. Let…

Let $X$ be a smooth projective $3$-fold over $\mathbb{C}$ and let $S\subset X$ be a nonsingular divisor. Let $I_0(X/S,n)$ be the moduli space of relative ideal sheaves, define … an…

Donaldson–Thomas degree-zero partition-function conjecture.

Let $X$) be a smooth projective complex variety. Let $\mathfrak I_n(X,\beta)$ be the moduli space of $1$-dimensional closed subschemes $Z$ of $X$ satisfying the paper's fixed numer…

Let $S\subset Y$ be a nonsingular, $T$-invariant divisor, let $\beta\in H_2(Y,\mathbb Z)$ satisfy $\int_\beta[S]\geq 0$, and let $\eta$ be a partition of $\int_\beta[S]$ weighted b…

Let $X$ be a smooth projective threefold, let $\overline{\mathcal{M}}_{g,n}(X;\beta)$ be the moduli space of stable maps, and let $\tau_k(pt)$ denote the Gromov–Witten descendent i…